Evolution of Tiling-like Crack Patterns in Maturing Columnar Joints

Fracture or cracking essentially involves the formation of new interfaces. These patterns are usually studied as two-dimensional mosaics. The new surface that opens up is in the third dimension, along the thickness of the sample. The thickness is usually very small compared to the lateral dimensions of the pattern. A spectacular and distinctive departure from these everyday examples of cracks are columnar joints. Here, molten volcanic lava, by the sea, cools and cracks under appropriate thermal and elastic conditions, causing the crack system to grow downward, creating long, vertical columns with polygonal cross-section. The focus of this paper is the study of the elongated interfaces of these columns: how the cross-section of their outlines gradually undergoes a metamorphosis from a disordered-looking Gilbert tessellation to a well-ordered hexagonal Voronoi pattern. As the columns grow downward to lengths of several meters (in natural systems), their outline continuously changes, the center may shift, causing the column to twist. For the first time, the evolution of these crack mosaics has been simulated and mapped as a trajectory of a 4-vector tuple in a geometry-topology domain. The trajectory of the columnar joint systems is found to depend on the crack seed distribution and crack orientation. An empirical relationship between the system energy and the crack mosaic shape parameter λ has been proposed on the basis of principles of fracture mechanics. The total system energy shows a power-law dependence on λ with the exponent β ∼ 0.3 and λ ≈ 0.75 at crack maturation. The parameter values are validated by matching the proposed relation with energy estimates existing in the literature. The relation not only matches the visible changes in geometry but also provides a feasible measure of the energy of the system. The geometric energy for the polygonal mosaics in the transverse section has also been estimated as a function of time. The geometric energy moves toward a minimum as the mosaic becomes more Voronoi-like at maturation.